A coin bank contains only nickels,dimes, and quarters. the value of the 19 coins in the bank is $2.00. There are twice as many nickels as dimes. Find the number of each type of coin in the bank.
How do you solve this?
1 25 25 5 50 1 1 10 10
To solve this problem, we can set up a system of equations to represent the given information.
Let's assume the number of nickels, dimes, and quarters in the bank to be N, D, and Q, respectively.
We are given three pieces of information:
1. The total number of coins is 19:
N + D + Q = 19
2. The value of the coins is $2.00:
0.05N + 0.10D + 0.25Q = 2.00
3. There are twice as many nickels as dimes:
N = 2D
Now, we can use these equations to solve for the values of N, D, and Q.
First, substitute the value of N in terms of D from equation 3 into equation 1:
(2D) + D + Q = 19
3D + Q = 19 [Equation 4]
Next, substitute the value of N in terms of D from equation 3 into equation 2:
0.05(2D) + 0.10D + 0.25Q = 2
0.10D + 0.10D + 0.25Q = 2
0.20D + 0.25Q = 2 [Equation 5]
Now, we have a system of two equations (equations 4 and 5) with two variables (D and Q). We can solve this system using substitution or elimination method.
Let's use the elimination method to solve for D and Q. Multiply equation 5 by 15 to eliminate the decimals:
3D + 3.75Q = 30 [Equation 6]
Now, subtract equation 4 from equation 6 to eliminate D:
(3D + 3.75Q) - (3D + Q) = 30 - 19
2.75Q = 11
Q = 11 / 2.75
Q = 4
So, we have found that the number of quarters is 4.
Now, substitute the value of Q = 4 into equation 4 to find the value of D:
3D + 4 = 19
3D = 15
D = 15 / 3
D = 5
So, we have found that the number of dimes is 5.
Finally, substitute the values of D = 5 and Q = 4 into equation 3 to find the value of N:
N = 2D
N = 2 * 5
N = 10
So, we have found that the number of nickels is 10.
Therefore, the number of each type of coin in the bank is:
Nickels: 10
Dimes: 5
Quarters: 4